Supersingular zeros of divisor polynomials of elliptic curves of prime conductor

Matija Kazalicki,Daniel Kohen

doi:10.1186/s40687-017-0099-8

Abstract

For a prime number p, we study the zeros modulo p of divisor polynomials of rational elliptic curves E of conductor p. Ono (CBMS regional conference series in mathematics, 2003, vol 102, p. 118) made the observation that these zeros are often j-invariants of supersingular elliptic curves over {overline{{mathbb {F}}_{p}}}. We show that these supersingular zeros are in bijection with zeros modulo p of an associated quaternionic modular form v_E. This allows us to prove that if the root number of E is -1 then all supersingular j-invariants of elliptic curves defined over {mathbb {F}}_{p} are zeros of the corresponding divisor polynomial. If the root number is 1, we study the discrepancy between rank 0 and higher rank elliptic curves, as in the latter case the amount of supersingular zeros in {mathbb {F}}_p seems to be larger. In order to partially explain this phenomenon, we conjecture that when E has positive rank the values of the coefficients of v_E corresponding to supersingular elliptic curves defined over {mathbb {F}}_p are even. We prove this conjecture in the case when the discriminant of E is positive, and obtain several other results that are of independent interest.

Highlights

Let E be a rational elliptic curve of prime conductor p
Denote by fE(τ ) ∈ S2( 0(p)) the newform associated with E by the Shimura–Taniyama correspondence
Serre [11, Theorem 11] showed that there is an isomorphism between modular forms modulo p of weight p + 1 and level 1 and modular forms modulo p of weight 2 and level p

Summary

Background

Let E be a rational elliptic curve of prime conductor p. The key idea to study these questions is to show (following [13]) how to associate with FE a modular form vE on the quaternion algebra B over Q ramified at p and ∞ Such modular form is a function on the (finite) set of isomorphism classes of supersingular elliptic curves over Fp. In order to explain this precisely, we combine the expositions from [3, 4]. We will show that the quaternion modular form vE associated with an elliptic curve E of rank 2 must be orthogonal to divisors arising from optimal embeddings of certain imaginary quadratic fields into maximal orders of the quaternion algebra B, leading to a larger amount of supersingular zeros

Proof of the main theorems

Modular forms on quaternion algebras

Proof of Theorem 4

Further remarks